Please answer ALL parts This is for a course in Probability Theory 42) Computational Problem Here is a probabilistic method for computing the area of a given subset S of the unit square The method uses a sequence of independent random selections of points in the unit square 0,1 x 0,1 , according to a uniform probability law if the ith point belongs to the subset S the value of a random variable Xi is set to 1, and otherwise it is set to 0 let x1, x2, be the sequence of random variables thus defined, and for any n, let Sn a) Show that E Sn is equal to the area of the subset S, and that var(Sn) diminishes to 0 as n increases b) Show that to calculate Sn, it is sufficient to know Sn 1 and Xn, so the past values of Xk, k 1, , n 1, do not need to be remembered Give a formula c) Write a computer program to generate Sn for n 1, 2, , 10000, using the computer's random number generator, for the case where the subset S is the circle inscribed within the unit square How can you use your program to measure experimentally the value of d) Use a similar computer program to calculate approximately the area of the set of all (x,y) that lie within the unit square and satisfy 0 cos x sin y 1 Problem 42 Computational problem Here is a probabilistic method for com puting the area of a given subset S of the unit square The method uses a sequence of independent random selections of points in the unit square 0, 1 0, 1 , according to a uniform probability law If the ith point belongs to the subset S the value of a random variable Xi is set to 1, and otherwise it is set to 0 Let X1, X2, be the sequence of random variables thus defined, and for any n, let TL Tt (a) Show that E Sn is equal to the area of the subset S, and that var(Sn) diminishes to 0 as n increases (b) Show that to calculate Sn, it is sufficient to know Sn 1 and Xn, so the past values of Xk, k 1, ,n 1, do not need to be remembered Give a formula , 10000, using the computer's random number generator, for the case where the subset S is the circle inscribed within the unit square How can you use your program to measure (c) Write a computer program to generate Sn for n 1,2, experimentally the value of (d) Use a similar computer program to calculate approximately the area of the set 1 of all (x,y) that lie within the unit square and satisfy 0

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Please answer ALL parts! This is for a course in Probability Theory. 42) Computational Problem. Here is a probabilistic method for computing the area of

image text in transcribed Please answer ALL parts! This is for a course in Probability Theory.

42) Computational Problem. Here is a probabilistic method for computing the area of a given subset S of the unit square. The method uses a sequence of independent random selections of points in the unit square [0,1] x [0,1], according to a uniform probability law. if the ith point belongs to the subset S the value of a random variable Xi is set to 1, and otherwise it is set to 0. let x1, x2, ... be the sequence of random variables thus defined, and for any n, let

Sn = image text in transcribed .

a) Show that E[Sn] is equal to the area of the subset S, and that var(Sn) diminishes to 0 as n increases.

b) Show that to calculate Sn, it is sufficient to know Sn-1 and Xn, so the past values of Xk, k=1, ... , n-1, do not need to be remembered. Give a formula.

c) Write a computer program to generate Sn for n = 1, 2, ..., 10000, using the computer's random number generator, for the case where the subset S is the circle inscribed within the unit square. How can you use your program to measure experimentally the value of image text in transcribed ?

d) Use a similar computer program to calculate approximately the area of the set of all (x,y) that lie within the unit square and satisfy 0 image text in transcribed cosx + siny 1.

Problem 42. Computational problem. Here is a probabilistic method for com- puting the area of a given subset S of the unit square. The method uses a sequence of independent random selections of points in the unit square [0, 1] [0, 1], according to a uniform probability law. If the ith point belongs to the subset S the value of a random variable Xi is set to 1, and otherwise it is set to 0. Let X1, X2,... be the sequence of random variables thus defined, and for any n, let TL Tt (a) Show that E[Sn] is equal to the area of the subset S, and that var(Sn) diminishes to 0 as n increases. (b) Show that to calculate Sn, it is sufficient to know Sn-1 and Xn, so the past values of Xk, k-1,...,n - 1, do not need to be remembered. Give a formula. ., 10000, using the computer's random number generator, for the case where the subset S is the circle inscribed within the unit square. How can you use your program to measure (c) Write a computer program to generate Sn for n -1,2, experimentally the value of ? (d) Use a similar computer program to calculate approximately the area of the set 1. of all (x,y) that lie within the unit square and satisfy 0